Theoretical investigations of molecular triple ionization spectra

Transcription

1 Theoretical investigations of molecular triple ionization spectra G. Handke, F. Tarantelli, and A. Sgamellotti Dipartimento di Chimica, Università di Perugia, Via Elce di Sotto 8, I Perugia, Italy L. S. Cederbaum Theoretische Chemie, Universität Heidelberg, Im Neuenheimer Feld 253, D Heidelberg, Germany Received 16 January 1996; accepted 5 March 1996 Triple ionization of molecular systems is investigated theoretically by means of the three-particle propagator. This enables us to efficiently calculate the very dense triple ionization spectra. To be able to interpret these spectra an atomic three-hole population analysis is developed which provides information about the charge distribution in the molecular trication. In exemplary applications on CO and a series of fluorides the use of the approach is demonstrated. A large number of triply ionized electronic states are energetically accessible in the available particle impact ionization experiments and it is shown that many of these states contribute to the observations. Triply ionized states are also produced by Auger decay. In particular, shake-off satellite lines of molecular Auger spectra can be reproduced using the triple ionization energies from the propagator calculation and an estimate of the transition rates based on the three-hole population analysis. In general a dramatically growing complexity of the triple ionization spectra with increasing molecular size is demonstrated. In spite of this complexity the three-hole population analysis is of valuable help for the interpretation of the spectra and often a simple picture in terms of various hole-localization patterns arises American Institute of Physics. S I. INTRODUCTION Processes which result in multiply charged molecular cations have gained growing importance in recent years. The multiple ionization of molecules is an interesting example of a transition process within a many-particle system which, because of the strength of correlation effects, often cannot a priori be understood in a simple independent particle picture. The study of the multiple ionization of molecules can provide useful information, for example, about the electronic structure, correlation effects, and the nature of the chemical bond. Triply charged cations represent the final states of many electronic processes and therefore often play a relevant part in ionization experiments. 1 7 Triply ionized molecules can be produced in various experimental arrangements. Multiple photoionization 1 became possible with the availability of powerful photon sources like synchrotrons or intense lasers. From particle accelerators beams of fast heavy ions are obtained which are used to study multiple ionization of molecules. 2 5 Furthermore also electron impact on molecules can lead to multiply ionized final states. 6,7 Because of the strong repulsion of the positive charges, the removal of three or more electrons from a molecule usually leads to a fragmentation or coulomb explosion of the resulting cation. Sophisticated experiments, in which some or even all fragments of the dissociation after ionization are detected in coincidence, give a detailed picture of these processes. Another type of experiment which involves triply ionized final states is Auger spectroscopy. 8,9 For example, if simultaneously with the creation of the core hole a second electron has been ionized in the initial state of the Auger transition, the final states after decay are mostly triply ionized. From the shake-off processes Auger satellite lines can arise in the experimentally obtained spectra. The assignment of these lines can be of fundamental importance for a correct interpretation of the resulting spectra. The experimental progress made in recent years is not yet sufficiently counterbalanced by accurate theoretical studies which can, besides their relevance by themselves, be of great help for the correct interpretation of the experimental results. Because of the enormous number of tricationic states and possible strong correlation changes and charge localization effects after the loss of three electrons, ab initio calculations of triply ionized molecular systems are quite involved. Conventional independent particle methods such as self-consistent field calculations yield results of unsatisfactory accuracy. More sophisticated methods, like configuration-interaction CI calculations, rapidly become prohibitively expensive for moderately large systems. Less conventional methods based on Green s functions, 10 which allow for the direct calculation of the transition observables without resorting to separate calculations for the initial and final states, can therefore be viewed as suitable instruments for the theoretical investigation of multiply charged cations. They have been successfully applied in the study of single 11 and double 9,12 14 ionization of molecules. The quantity suitable for the analysis of triple ionization processes within the Green s functions approach is the three-particle propagator. In this work we report on the use of the second-order algebraic diagrammatic construction ADC approximation of this quantity for the study of the triple ionization of molecules. In Sec. II we describe the J. Chem. Phys. 104 (23), 15 June /96/104(23)/9531/15/$ American Institute of Physics 9531

2 9532 Handke et al.: Molecular triple ionization spectra theoretical basis and some technical facts of our method. This includes the implementation of the second-order working equations given in Ref. 18 and the development of the atomic three-hole population analysis, which allows the investigation of space localization of electron vacancies and of the charge distribution in the dense manifold of trication states. Section III contains exemplary applications which demonstrate that the presented method is very suitable for the theoretical study of triply ionized molecular systems and is helpful in the interpretation of experimental findings. II. THEORY In this section we briefly review the theoretical basis of our method, starting with some facts about the three-particle Green s function, which are of relevance for the present work. Then the ADC approximation of the three-particle propagator and the implementation of the second-order ADC working equations for this propagator, which allow the efficient calculation of a large number of triply ionized states, are discussed. Finally we report on the population analysis of the Green s function three-hole pole strengths which gives, for each triply ionized state, a measure of the spatial distribution of charges over the trication. A. Three-particle propagator The three-particle Green s function in its general form is given by a time-ordered expectation value of three creation and three annihilation operators, 18 G, i 3 0 N T a t a t a t a t a t a t N 0, 1 where N 0 is the exact N-particle ground state and T the Wick s time-ordering operator. 10 The creation (a ) and annihilation (a ) operators in the Heisenberg representation are related to a suitable basis of single-particle states. In dealing with molecular systems these states are commonly chosen as the molecular orbitals from a self-consistent field calculation for the ground state of the neutral molecule. The three-particle Green s function can be divided into different parts, one of which describes the transition processes from the N-particle system to the (N 3)-particle system. A particular choice of time arguments leads to that part of the three-particle propagator which describes the simultaneous ejection of three particles from the N-particle ground state. For a time-independent Hamiltonian the Fourier transform yields the following spectral representation of the quantity of interest for details see Ref. 18 :, N 0 a a a N 3 m N 3 m a a a N 0 m E N N E m Here (N 3) m is the complete set of eigenstates of the Hamiltonian in the (N 3)-particle space. The triple ionization spectrum appears explicitly in expression 2. The poles of : m E 0 N E m N 3, give the triple ionization potentials TIPs, i.e., the energy differences between the (N 3)-particle states m (N 3) and the N-particle ground state 0 N. The corresponding transition amplitudes are represented by the residues 3 x m N 3 m a a a N 0. 4 Using the fact that the set m N 3 of (N 3)-particle states is complete, Eq. 2 can be rewritten without referring to a specific representation of the (N 3)-particle space, 0 N a a a E 0 N Ĥ 1 a a a 0 N. Here Ĥ is the Hamiltonian of the system. B. ADC approximation Equation 5 is the starting point for the derivation of a second-order approximation scheme for which has been carried out in Ref. 18. Two different methods, the ADC approximation 15,16 and a purely algebraic approach, 19 were shown to yield equivalent working equations represented in different forms. Our implementation is based on the working equations as they are obtained from the ADC approach, which will be briefly described in the following. The ADC method is a general constructive approach to the derivation of order-by-order terms in the perturbation expansion of the Green s function. It was first applied to the polarization propagator 15 and to the one-particle Green s function. 16 A particular nondiagonal representation of the propagator is the central point of the ADC approach. Recently a closed form expression for the transformation between this ADC representation and the diagonal representation has been derived. 17 The three-particle propagator is expressed in the following nondiagonal representation of the general expression in Eq. 5 : f K C 1 f. The matrix K is the diagonal matrix of zeroth-order energy differences, C is an effective interaction matrix, and f is the matrix of effective transition amplitudes. The configuration space of these matrices is the (N 3)-particle space. An approximation for the three-particle propagator is obtained by expanding Eq. 5 in power series of the matrix K 1 C using perturbation expansions of the matrices f and C. By comparing order by order this expansion with the corresponding diagrammatic expansion in terms of Goldstone diagrams the matrix elements of the matrices K, C, and f are derived in Ref. 18 up to and including the second order of perturbation. For a given approximation of these matrices the inversion problem of Eq. 5 is equivalent to the eigenvalue problem 15,16 5 6

3 Handke et al.: Molecular triple ionization spectra 9533 K C X X. Here is the diagonal matrix of the eigenvalues, which are the TIPs of the system under investigation, and the transition amplitude to the th triply ionized state, described by the eigenvector X, is given by f X. The nth-order ADC scheme, which is given by the expansion of C and f up to nth order, represents an infinite partial summation for the diagrammatic perturbation expansion being exact up to nth-order perturbation theory. In the ADC approach the required configuration space is smaller than that of a comparable CI expansion and grows considerably slower with increasing order n. For n 2 and n 2 1 the ADC configuration space comprises the 1 lowest classes of excitation. In the second-order ADC scheme for triple ionization this means all three-hole (3h) and four-hole one-particle (4h1p) configurations. A comparable CI expansion also requires 5h2p configurations. Of particular relevance for the applications to large systems is that the ADC scheme is size consistent in each order. With the second-order ADC working equations given in Ref. 18 the main states, perturbatively derived from 3h configurations, are treated consistently at second order of perturbation theory but terms to infinite order are included in a systematic way. The secondary states are treated consistently at first order, again including systematically terms to infinite order. To build up the corresponding ADC matrix based on Hartree Fock HF spatial orbitals these equations, which were given in spin orbitals, are for convenience transformed into a spin-free notation. This is done by writing each configuration as a product of a spatial part and a spin eigenfunction. Starting from a closed shell N-particle ground state there are up to ten spin eigenfunctions for a 4h1p configuration of the trication, one sextet, four quartets, and five doublets. For each multiplicity and each molecular symmetry there is a separate ADC matrix. Because the sextet states arise only from the 4h1p space these are not computed. Each 3h and 4h1p configuration of HF spatial orbitals gives rise to a small block up to 5 5 for doublet configurations with five singly occupied spatial orbitals of the ADC matrix. Only for very small systems can the resulting ADC matrices be fully diagonalized by standard routines. For larger matrices we use carefully optimized diagonalization routines based on block Lanczos 20,21 or block Davidson 21 algorithms to extract a large number of eigenvectors and eigenvalues. 7 C. Population analysis To investigate the distribution of charges in the dense manifold of the triply ionized final states and the space localization of the electron vacancies we use a Mulliken-like population analysis 22 of the 3h pole strength. This kind of analysis was successfully applied to doubly ionized states and it was shown to be very useful in the theoretical simulation of Auger spectra and in particular in gaining deeper insight into the influence of charge distribution and localization effects on these spectra In the following we describe the extension of this method to the 3h population analysis of tricationic states. To obtain a measure of the spatial distribution of charges over the trication for each triply ionized state the 3h component of the corresponding eigenvector here this will be the ADC eigenvector of the triply ionized state is analyzed in terms of the atomic orbital AO basis. This is done by first expanding the spin adapted 3h configuration in the AO basis: ijk d,q d,q U prs,ijk p r s p,s prs d,q d U pps,ijk pps d. The superscripts d and q denote doublet or quartet spin multiplicity, respectively, i, j, k are hole indices, and p, r, and s label AO basis functions. For a configuration pps there is no quartet state, therefore the second sum in Eq. 8 is zero for quartet states. For i j k there are two doublet spin functions and the left-hand side of Eq. 8 is a two-component vector containing the two 3h configurations with the same spatial orbitals and different spin functions. In this case (d) U prs,ijk (p r s)isa2 2 matrix. The matrix elements of U (d) and U (q) can be expressed in terms of the HF eigenvector matrix C. Using the expansion 8 i p c pi p 9 we obtain, for example, for i j k and p r s d U prs,ijk 3 ijk rps ijk srp ijk rps ijk srp ijk rps ijk srp ijk prs 1 ijk rps srp ijk 1 ijk prs 3 10 q U prs,ijk ijk prs ijk rps ijk srp,

4 9534 Handke et al.: Molecular triple ionization spectra with ijk prs c pi c rj c sk c sj c rk, ijk prs c pi c rj c sk c sj c rk. 11 Similar expressions are obtained for the other cases i j and/or p r which are not reported here. The orthonormality condition of the molecular orbital functions gives U OU 1, 12 where O is the overlap matrix over the AO functions. O can be expressed in terms of the basis set overlap matrix S. We give here, as an example, the expression for quartet configurations p r s, t u v, q O prs,tuv S pt S ru S sv S rv S su S pu S rv S st S rt S sv S pv S rt S su S ru S st. 13 Again, the similar expressions for the other cases are not given here. A generic eigenvector X of the th tricationic state can be represented in terms of AO functions: Y UX. The total 3h weight of this eigenvector is now given by X X Y OY prs Q prs,, with Q prs, Y prs, tuv Y tuv, O prs,tuv. 16 The summation in the last two equations is over all possible 3h functions, i.e., p r s and p r s. As already mentioned above, X is the ADC eigenvector in our approach. Equation 16 provides a general, well-defined way to analyze the tricationic states in terms of atomic contributions. Q prs, is interpreted as the contribution of the function prs to the total 3h weight of the th state and the sum of all terms Q prs, where prs is characterized by the same localization pattern which gives the pole strength for this pattern. With the term localization pattern we refer to the various possibilities of localization of the three holes at different atomic sites. These are the one-site terms, with all three holes at the same atomic site, the two-site and the three-site terms with the holes located at two or three different atomic sites, respectively. For a three-atomic molecule XYZ the one-site terms are X 3, Y 3, and Z 3, and the one-site pole strength of the atom X is the sum of all terms Q prs,, where p, r, and s refer to basis functions centered on X. Similarly, the two-site pole strength X 2 Y 1 is the sum of all terms with two basis functions centered on X and one centered on Y, and the three-site pole strength X 1 Y 1 Z 1 is the sum of all terms where each of the three atomic basis functions is centered on a different atom. For the actual implementation Eq. 16 is rewritten in a different form to allow a more efficient computation and to avoid storage of the possibly huge matrix U: Q prs, lmn U tuv,lmn X lmn, ijk ijk ijk U prs,ijk X ijk, tuv O prs,tuv lmn X ijk, X lmn, U prs,ijk tuv O prs,tuv U tuv,lmn prs X ijk, X lmn, W ijk,lmn, 17 lmn where the last equation defines the matrix W. This matrix has the dimension of the number of 3h states, which is usually relatively small compared to the full dimension of the configuration space, and can be expressed in terms of the matrix elements of the HF eigenvector matrix C and the basis set overlap matrix S. InEq. 17 there is a matrix W for each configuration prs, but W is the only quantity which depends on the basis set configuration prs and we are only interested in the sums over all contributions with the same localization pattern. Therefore we can carry out the summation over all configurations with the same localization pattern in Eq. 17 before multiplying with the eigenvectors X and only few relatively small matrices as many as the number of different localization patterns are needed to carry out the three-hole population analysis for all eigenvectors. III. APPLICATIONS The main purpose of the present work is the application of the theoretical methods described above to the comparative study of the valence triple ionization spectra of a series of related molecules. Given the possibility of computing vast portions of these spectra opened up by the Green s function method, a relevant subject of investigation appears to be the charge distribution patterns one may observe in the dense tricationic manifolds, how these affect the energy distribution of the states, and how they vary with related chemical environments. Ionic systems where hole localization effects may be expected to be particularly pronounced, and thus where simple physical models may guide our learning, appear to be appropriate examples in this respect and we have chosen to study the series of fluorides LiF, BeF 2, and BF 3. As mentioned in Sec. I, the theoretical study of triple ionization spectra is of valuable help for the understanding of a variety of experimental observations where trications are directly or indirectly measured. For completeness, before discussing the main results of the present application, we begin this section by briefly reviewing two previous studies that illustrate some of these aspects of our work. A. Energy distribution of triply ionized states, CO In our first example we discuss the importance of taking into account a very large number of triply ionized states in the interpretation of triple ionization experiments very

5 Handke et al.: Molecular triple ionization spectra 9535 FIG. 1. Electron impact total triple ionization cross section of carbon monoxide. The solid line is the theoretical result obtained by giving each state the same, energy independent, triple ionization probability. For the dashed line a 50 times higher triple ionization probability is assumed for the 3h configurations with respect to the 4h1p configurations. The diamonds are the experimental results Ref. 7 obtained by summing the data for the two dissociation channels. likely, the corresponding is also true for experiments involving even higher ionized states. Triple ionization of molecules generally leads to the dissociation of the resulting trication and the two or more fragments are often observed in coincidence. The measured quantities are the masses and charges of the fragments and their kinetic energies. The experimental findings 1 6 are usually interpreted by calculating the potential energy curves of a few energetically low lying dissociating states or by adopting a Coulomb explosion model in which the individual fragments are treated like point charges repelling each other by the Coulomb interaction. Because the density of the tricationic states is considerably high even for quite small molecules a large number of states are accessible in these experiments. In the following we shall demonstrate, by example of the triple ionization of CO, that this large number of states actually contributes to the experimental results and therefore their interpretation cannot safely be based on the consideration of only a few tricationic states. In a recent coincidence measurement of the dissociation products of CO 3, obtained by electron impact, the triple ionization cross section has been determined. 7 It has been found 23 that this cross section can be reproduced quite satisfactorily using only the energy distribution of the TIPs calculated by the ADC method. In the very simple model, that each tricationic state contributes to the cross section with the same, energy-independent value above the corresponding triple ionization threshold, the triple ionization cross section is given by (E) N(E)/ (E), where N(E) is the integrated density of states and E is the impact energy. 23 This very crude approximation of equal and energy-independent contributions to the cross section is certainly not correct for any individual state but is assumed to be reasonable as an average over a large number of states. Figure 1 shows the theoretical triple ionization cross section computed with N(E) obtained from an ADC calculation in a triple-zeta valence [5s,3p] contracted Gaussian basis set augmented with one d function on each atom 24 using the equilibrium bond distance of the CO ground state. 25 For comparison the experimental result the sum of the two dissociation channels C 2 O and O O 2 of Spekowius and Brehm 7 is also shown in Fig. 1. A second theoretical curve which accounts for the higher probability to produce states dominated by the 3h configuration 23 is also given. The fact that our theoretical results correspond nicely to the experimental cross section suggests that the shape of the cross section is essentially dictated by the form of the dense energy distribution of triply ionized states. This implies that not only a few low lying tricationic states but a very large number of them must be populated in the ionization process. From direct triple ionization experiments also more detailed information on the actual dynamics of fragmentation of the molecular trications is obtained. Of course, to interpret these results the calculation of vertical triple ionization transitions does not suffice, as the nuclear dynamics on the enormous number of populated potential energy surfaces is involved. Our calculations do, however, indirectly provide some useful information. The population analysis of the triply ionized states of CO after vertical ionization shows 23 that the three holes in essentially all states are largely delocalized, whereas all the experiments consistently indicate a large predominance of the C 2 O dissociation channel over the C O 2 one. This is a clear indication of strong nonadiabatic coupling among the densely packed potential energy curves on which the nuclear dynamics of dissociation proceeds. B. Shake-off satellites in Auger spectra, LiF Our next example is dedicated to molecular Auger spectra and in particular to the shake-off satellite lines of these. Shake-off or shake-up satellite lines arise if in the intermediate core hole state, which is decaying via an Auger process, a valence electron has also been ionized or excited, respectively. These satellite lines can carry a significant portion of the Auger intensity so that their correct assignment becomes crucial for the interpretation of the spectrum. Triply ionized states are the final states of shake-off transitions and beside the sufficiently accurate calculation of the TIPs an estimate of the intensities is needed for a theoretical reproduction of the spectrum. In principle, the continuum wave that describes the emitted Auger electron must be computed to obtain the Auger transition rates. 26 This is quite involved and up to now is feasible only for rather small systems. On the other hand, the large number of final states not only makes the accurate calculation of individual transition rates very difficult or even impossible but also allows a statistical approach 27 which is feasible for large molecules. For the normal Auger transitions to dicationic final states it has been shown that this method together with an atomic population analysis 22 gives satisfactory estimates of the relative intensity distributions. Because the triply ionized states are

6 9536 Handke et al.: Molecular triple ionization spectra FIG. 2. The experimental upper Ref. 28 and theoretical lower Auger spectrum of LiF. The individual transitions are indicated by vertical bars with solid lines for transitions to doubly ionized final states normal Auger and dotted lines for transitions to triply ionized final states shake-off. The heights of the bars correspond to the intensities of the transitions which are given in arbitrary units. To obtain the theoretical spectrum the intensities of the individual transitions are convoluted with Gaussian functions of full width at half-maximum of 1.0 ev for transitions to dicationic final states and 2.0 ev for transitions to tricationic final states. A triplet singlet population ratio of 1.36 Ref. 32 is used for the intermediate states of the shake-off transitions. even more numerous than the dicationic states this approach is well suited for the calculation of shake-off contributions to molecular Auger spectra. The Auger spectrum of LiF in gas phase has been measured by means of electron-beam excitation. 28 It shows several satellite bands and the integrated Auger satellite intensity of this spectrum was found to amount to 25%. 28 Using the triple ionization energies calculated by the ADC method and the estimate of the intensity distribution based on a three-hole population analysis the satellite lines of the Auger spectrum of LiF can be reproduced. 29 Figure 2 shows the theoretically simulated spectrum together with the experimental spectrum of Hotokka et al. 28 For the theoretical spectrum the double and triple ionization potentials have been computed by the second-order ADC approximation in a [4s,2p] contracted Gaussian basis set augmented with a 1d polarization function double zeta polarization basis, DZP 30 and the experimental equilibrium distance 25 of the neutral LiF ground state has been used. The intensities of the normal Auger transitions have been taken from the calculation of Zähringer et al. 31 For the shake-off transitions four intermediate decaying states with a core and a valence hole have been considered which are assumed to be nonstatistically populated with the triplet singlet population ratio from the similar neon atom of The intensity distribution for these transitions has been estimated by attributing to each transition a contribution proportional to the F 3 one-site pole strength. All these contributions have been adjusted by one common factor to account for the lower probability of the shake-off transitions compared to the normal Auger transitions. The individual transitions, indicated in Fig. 2 by vertical bars of height corresponding to their estimated intensity, are convoluted by Gaussian functions to obtain the theoretical spectrum. The convolution accounts qualitatively for the broadening of the bands due to nuclear dynamics. An accurate estimate of nuclear dynamics effects due to the finite lifetime of the decaying states, beyond the scope of this first application, could be achieved by the methods developed in Ref. 33. It can be seen from Fig. 2 that the overall agreement between theory and experiment is satisfactory. The previously unassigned experimental feature corresponding to peak 8 of the theoretical spectrum is nicely reproduced. This example shows that the calculation of shake-off contributions to molecular Auger spectra with the method described here is of sufficient accuracy to be of valuable help in the interpretation of experimental spectra and yet requires such a moderate computational effort that it is applicable to not only the smallest molecular systems. C. The fluorides LiF, BeF 2, and BF 3 We now turn to our main application in this work, the investigation of the triple ionization of the fluorides LiF, BeF 2, and BF 3, comparing the results for the individual molecules with each other. The choice of these fluorides appears at the outset to be an appropriately simple example in order to familiarize with and establish guidelines for the study of triple ionization in molecules: these three molecules are ionic and hole localization effects are expected to be clearly visible. The analysis of the double ionization spectra of polyfluorides has demonstrated 13,22 that the two-electron vacancies in the dicationic states are strongly localized around the fluorine atoms, giving rise to two main classes of states characterized as two-site, where each of a pair of fluorine atoms carry one hole, and one-site, where both vacancies are localized mostly on a single fluorine. Some remarkable consequences of this charge distribution character on the clustering in energy and spectroscopic properties of the dicationic states have been studied in detail. 13,22 The additional hole present in triply ionized states enriches this picture giving rise to more charge distribution patterns, depending on the number of fluorine atoms present, and to different interplays among them. We would like to establish if and when the presence of a third vacancy induces delocalization of the charges and if, at the other extreme, localization of all three holes at the same electronegative site may take place. The latter one-site localization would then characterize the principal class of states in LiF. In BeF 2 the additional class of two-site states, with two holes localized at one fluorine and one hole at the other, should be observed. Finally, in BF 3, three-site states each fluorine carrying one vacancy also enter the picture. It is also very interesting to analyze to what extent in the triply ionized states, or parts of their spectra, with respect to the double ionization case, the charge distribution affects the central atom. All calculations reported in this section have been carried out using the DZP basis set and the geometry of the respective neutral ground states. The BeF 2 molecule is linear with symmetry D h and BF 3 is planar with symmetry D 3h.

7 Handke et al.: Molecular triple ionization spectra 9537 FIG. 3. Calculated integrated density N(E) of triply ionized states for LiF, BeF 2, and BF 3. The number of states divided by the square root of the energy is scaled to reach the maximum height 1 for all three molecules. The experimental bond lengths of Å Li F, Å Be F, 34 and Å B F 35 have been used. The HF ground state configurations are LiF, 1 2 u 1 2 g 2 2 g 2 2 u 3 2 g 3 2 u 4 2 g u 1 g and BeF, core 1a 1 2 1e 4 2a 1 2 2e 4 1a 2 3e 4 1e 4 1a BF 3. In the ADC calculations the core orbitals, one for LiF, two for BeF 2, and four for BF 3 are kept doubly occupied. The resulting ADC matrices range in size from 269 to Up to a dimension of about 3200 these matrices have been fully diagonalized by standard routines; from the other matrices a large number of eigenvectors and eigenvalues have been obtained by a recently implemented block Lanczos method. 21 Because the amount of computed data on the vast number of triple ionization transitions is enormous, it is impossible to report here all results in detail. We have tabulated some exemplary data and concise summaries of our results. Together with the following discussion and the presented FIG. 5. The total 3h pole strength of the triply ionized states of BeF 2 is shown as a bar spectrum. figures the physical content of the data should be sufficiently enlightened. All data are available on request. In Fig. 3 the energy distribution of the calculated triply ionized states for each molecule is shown. As has been discussed above the quantity N(E)/ (E) can be taken as an approximation of the total triple ionization cross section and is therefore given in Fig. 3 instead of only the number of states N(E). All curves are scaled to a maximum height of 1 to enable comparison. The results for the three molecules are quite similar and also similar to the result for CO, see Fig. 1, with, however, visible differences in the onset, the position of the maximum, and the slope at low energies. To our knowledge, no experimental data are available for comparison. The distribution in energy of the total 3h weight can be seen in the Figs. 4 6, where for each tricationic state a vertical bar of height proportional to the total 3h pole strength is plotted at the corresponding triple ionization energy. One immediately notes the very rapidly growing number of states going from LiF to BF 3. The number of computed states in the lower part of the spectrum increases much faster than the dimension of the ADC matrices, which goes with the fourth power of the number of electrons in the system. In an energy FIG. 4. The total 3h pole strength of the triply ionized states of LiF is shown as a bar spectrum. FIG. 6. The total 3h pole strength of the triply ionized states of BF 3 is shown as a bar spectrum.

8 9538 Handke et al.: Molecular triple ionization spectra TABLE I. Results of the ADC calculation and the three-hole population analysis for triply ionized states of LiF. The first column designates the state, the second column shows the triple ionization potential TIP in ev. The third column gives the total three-hole pole strength and the columns four to seven contain its distribution to the different contributions of the population analysis. In the last column the square coefficients of the most important configurations in the ADC eigenvector are shown. The lowest 24 states are shown. State TIP Total Li 3 F 3 Li 1 F 2 Li 2 F 1 Configurations range of 80 ev above the triple ionization threshold we found 47, 543, and 8111 triply ionized states with a total three-hole pole strength larger than of LiF, BeF 2, and BF 3, respectively. Clearly, this enormous growth of the number of states is accompanied by an increasing complexity of the corresponding spectra. Whereas the distribution of the triply ionized states of LiF is readily understood from the three-hole configurations and the hole localization of the individual states, an interpretation of the BF 3 spectrum based on the three-hole configurations of the individual states is not appropriate, not only because of the sheer number of states but also because of the strong configuration mixing. 1. LiF The triply ionized states of LiF shown in Fig. see also Table I can be divided in five groups which correspond to different 3h configurations. Between 80 and 90 ev triple ionization energy the states are characterized by three holes in the two outermost orbitals 1 and 4. In this group the state with the lowest TIP is a and about 4 ev higher in energy the doublets appear. The second group from 99 to 115 ev ionization energy comprises the states with one hole in the 3 orbital and the remaining two holes in one or both of the two outermost orbitals. Here, as a consequence of the smaller distance in space of the three holes, the difference between quartets and doublets is larger and amounts to about 7 ev. All three orbitals which are involved in the ionization in the first two groups are well localized on the flourine atom, so that a localization of the three holes at this atom can be expected. This is confirmed by the three-hole population analysis, as can be seen in Table I. In the third group with TIPs between 119 and 127 ev we find states having one hole in the 2 orbital and again the other two holes in the two outermost orbitals. Here the states are characterized by a two-site Li 1 F 2 hole localization reflecting the localization of the 2 orbital on the Li atom. The fourth group is formed by only two states which have two holes in the 3 orbital and one hole in the 1 or 4 orbital, respectively, and therefore a localization of all three holes at the fluorine atom. In the last group above 140 ev there are again states of Li 1 F 2 type with one hole in the 2 orbital, one in the 3 orbital, and the remaining hole in one of the outermost orbitals. 2. BeF 2 Already more involved is the situation for BeF 2. In Table II the lowest 50 calculated states are shown and Table III summarizes results of the three-hole population analysis. In the latter the averages over groups of states with similar character are reported. Three large groups of main states can be discerned in Fig. 5, the first up to 81 ev triple ionization energy, the second from 93 to 106 ev, and the last between 114 and 131 ev. The composition of these groups, however, is not as homogeneous and easy to survey as it was in the case of the LiF molecule. One reason for this is the considerable mixing of 3h configurations. Even more important are the effects due to hole localization at the two equivalent fluorine atoms. The large hole hole repulsion leads to low-lying states which have two holes localized on one fluorine atom

9 Handke et al.: Molecular triple ionization spectra 9539 TABLE II. Computed triple ionization potentials TIPs in ev, total three-hole pole strength ps, and 3h composition of tricationic states of BeF 2. The composition is given by the square 3h components of the ADC eigenvectors larger than 0.1 ev. The 3h configurations are indicated by the orbitals which are occupied in the ground state of BeF 2 and from which three electrons are removed. The lowest 50 states are shown. State TIP ps Configuration u (1 2 g 1 1 u ) g (1 1 g 1 2 u )0.3526(1 3 g ) u (1 2 g 1 1 u )0.2545(1 3 u ) g (1 1 g 1 2 u ) g (1 2 g 4 1 g )0.1091(1 1 g 1 1 u 3 1 u ) u (1 2 g 3 1 u )0.1673(1 1 g 1 1 u 4 1 g ) u (1 1 g 1 1 u 4 1 g )0.2248(1 2 g 3 1 u ) u (1 1 g 1 1 u 4 1 g ) u (1 1 g 1 1 u 4 1 g ) g (1 1 g 1 1 u 3 1 u )0.3042(1 2 g 4 1 g ) g (1 2 g 4 1 g )0.2779(1 2 u 4 1 g )0.1820(1 1 g 1 1 u 3 1 u ) g (1 2 g 4 1 g )0.2944(1 2 u 4 1 g )0.1821(1 1 g 1 1 u 3 1 u ) u (1 2 g 3 1 u )0.2580(1 2 u 3 1 u )0.2252(1 1 g 1 1 u 4 1 g ) u (1 2 g 3 1 u )0.2735(1 2 u 3 1 u )0.2259(1 1 g 1 1 u 4 1 g ) g (1 1 g 1 1 u 3 1 u ) g (1 1 g 1 1 u 3 1 u ) u (1 1 g 1 1 u 4 1 g )0.2367(1 2 u 3 1 u )0.1068(1 2 g 3 1 u ) g (1 1 g 1 1 u 3 1 u )0.2725(1 2 u 4 1 g ) g (1 2 u 4 1 g )0.2911(1 1 g 1 1 u 3 1 u ) u (1 2 u 3 1 u )0.2439(1 1 g 1 1 u 4 1 g ) u (1 2 g 1 1 u ) g (1 1 g 1 2 u )0.1825(1 3 g ) u (1 2 g 1 1 u )0.2201(1 3 u ) u (1 1 g 4 1 g 3 1 u ) g (1 2 u 1 1 g ) g (4 2 g 1 1 g )0.2899(3 2 u 1 1 g )0.1652(1 1 u 4 1 g 3 1 u ) u (4 2 g 1 1 u )0.2510(3 2 u 1 1 u )0.2437(1 1 g 4 1 g 3 1 u ) g (1 1 u 4 1 g 3 1 u ) u (1 2 g 1 1 u )0.1217(1 3 u ) g (1 1 g 1 2 u ) u (1 1 g 1 1 u 4 1 g )0.2338(1 2 g 3 1 u ) u (1 1 g 1 1 u 4 1 g )0.1340(1 2 g 3 1 u ) g (1 2 u 4 1 g )0.2548(1 2 g 4 1 g )0.2244(1 1 g 1 1 u 3 1 u ) g (1 1 g 1 1 u 3 1 u )0.2306(1 2 g 4 1 g ) g (1 1 g 1 1 u 3 1 u )0.1082(1 2 g 4 1 g ) u (1 2 u 3 1 u )0.2749(1 2 g 3 1 u )0.1831(1 1 g 1 1 u 4 1 g ) u (1 1 g 1 1 u 4 1 g )0.3468(1 2 u 3 1 u ) g (1 1 g 1 1 u 3 1 u )0.3389(1 2 u 4 1 g ) u (1 1 g 1 1 u 4 1 g )0.2036(1 2 u 3 1 u )0.1574(4 2 g 3 1 u )0.1221(1 2 g 3 1 u ) g (1 1 g 1 1 u 3 1 u )0.2105(1 2 u 4 1 g )0.1398(3 2 u 4 1 g )0.1324(1 2 g 4 1 g ) u (1 1 g 4 1 g 3 1 u )0.1414(4 2 g 1 1 u )0.1057(3 2 u 1 1 u ) g (1 1 u 4 1 g 3 1 u ) u (1 1 g 4 1 g 3 1 u )0.1959(3 2 u 1 1 u )0.1363(4 2 g 1 1 u )0.1195(1 2 g 1 1 u ) g (1 1 u 4 1 g 3 1 u )0.1875(3 2 u 1 1 g )0.1413(4 2 g 1 1 g ) u (4 2 g 3 1 u ) g (3 2 u 4 1 g ) g (1 2 g 3 1 g )0.3037(1 2 u 3 1 g ) u (1 1 g 1 1 u 3 1 g ) u (1 1 g 1 1 u 3 1 g ) u (1 1 g 1 1 u 3 1 g )0.1133(1 2 g 2 1 u ) and one hole localized on the other and high-lying counterparts with three holes localized on the same atom. These symmetry-breaking effects cannot be described by symmetry-restricted Hartree Fock methods. 22 Because of the large energy differences between the one-site and the two-site states, tricationic states with corresponding 3h composition are found in different groups in the spectrum. By means of the three-hole population analysis the different characters of the states with corresponding 3h composition can be uncovered. The F 2 F 1 two-site and the F 3 one-site contributions to the states of the three groups mentioned above are shown in Fig. 7 see also Table III. There is no hole localization on the Be atom. It can be seen in Fig. 7 that the first states with one-site localization appear in the second group. These states correspond in their 3h composition to the states of the first group and are separated from them essentially by the hole hole repulsion see Sec. III C 4. The same relation can be found for the one-site localized states of the

10 9540 Handke et al.: Molecular triple ionization spectra TABLE III. Results of the three-hole population analysis for BeF 2. The average over groups of similar states is given and the overlapping energy ranges which comprise the states are indicated in the table. The first line for each group shows the average value of the different contributions and the second line the corresponding standard deviations. Energy inteval F 3 F 2 F 1 Be 3 Be 2 F 1 F 2 Be 1 F 1 F 1 Be 1 Total FIG. 7. The F 3 one-site and F 2 F 1 two-site contributions of the three-hole population to the triply ionized states of BeF 2. third group and the two-site localized states of the second group with a somewhat smaller separation in energy. The arrangement of the states with the same hole localization in three groups is due to the large energy difference of more than 23 ev between the 3 u and the 3 g orbital. All tricationic states of the first group are characterized by three holes in the four outermost orbitals. In the second group the twosite localized states have one hole in one of the energetically similar 3 g or 2 u orbitals. Correspondingly, in the third group, states with two holes in these orbitals are found. Very interestingly, because of the similar magnitude of the energy separation between states with different hole localization and states with different 3h composition, one-site and two-site localized states with different 3h composition are found in the same energy region. In these mixed groups the hole localization is less pronounced and the variation of the terms of the three-hole population analysis in groups of similar character is larger, which can be seen on the greater standard deviation in Table III for these groups. This underlines the general inherent complexity of the energy and character distribution of the triply ionized states, which can be usefully interpreted only by resorting to a charge distribution analysis as employed here. 3. BF 3 The tricationic states of BF 3 show a very strong mixing of 3h configurations already at low triple ionization energy. In Table IV the 65 energetically lowest and some of the higher-lying states are given together with their total 3h pole strength and 3h composition of the ADC eigenvector. All 3h configurations with a square coefficient greater than 0.1 are reported. There is only one state which is essentially described by a single 3h configuration, namely the 4 A 1 state with a triple ionization energy of 71.9 ev and the 3h configuration 1e 2 1a 1. Only very few states have a 3h composition dominated by a single configuration. For the vast majority of the states the largest contribution of a single configuration to the 3h composition is less than 50% of the total 3h pole strength and for a large number of states with a 3h pole strength greater than 0.1 this ratio is even less than 20%. One example of these is the 2 E state at 81.4 ev, which has a total 3h pole strength of and the largest square coefficient of 3h configurations is 0.13 of the configuration 1e 2 2a 1 1 ; all other 3h configurations contribute with square coefficients less than 0.1 and therefore are not shown in Table IV. Despite the strong mixing of the 3h configurations of nearly all tricationic states, the total 3h pole strengths of the states up to a TIP of about 95 ev show an evident regularity clearly visible in Fig. 6. There are essentially two different values of the total 3h pole strength of these states, about 0.85 for the lower-lying states and about 0.8 in the following. This uniformity of the pole strengths over wide energy ranges is a clear indication of a common physical character of the underlying tricationic states. From the three-hole population analysis we learn that these two types of states differ in the hole localization. The first type, with total pole strengths of about 0.85, is characterized by a three-site localization with all three holes at different fluorine atoms,

11 Handke et al.: Molecular triple ionization spectra 9541 TABLE IV. Same as Table II for triply ionized states of BF 3. The lowest 65 and some higher lying states are shown. State TIP ps Configuration 2 E (1a 2 3e 1 )0.1649(1a 1 3e 2 )0.1029(3e 3 ) 4 A (1a 1 3e 2 )0.2400(1a 1 3e 1 2e 1 ) 4 A (1a 1 1e 1 3e 1 )0.1242(1a 1 1e 1 2e 1 ) 2 E (1a 2 1e 1 )0.1094(3e 2 1e 1 ) 2 A (1a 1 1e 1 3e 1 )0.1315(1a 1 1e 1 2e 1 )0.1066(3e 2 1a 1 ) 4 E (1a 1 1e 1 3e 1 )0.1667(1a 1 3e 1 1a 1 )0.1589(1e 1 3e 2 ) 2 E (1a 1 1e 1 3e 1 )0.1798(1a 1 3e 1 1a 1 )0.1608(1e 1 3e 2 ) 4 A (1a 1 1e 2 )0.2882(1e 1 3e 1 1a 1 ) 2 A a 21a a 11e 13e e 13e 12e e 13e 2 1) E (1a 1 1e 2 )0.2229(1e 2 3e 1 )0.1183(1a 1 1e 1 1a 1 ) 2 A (1a 1 1e 2 )0.3228(1e 1 3e 1 1a 1 ) 4 E (1e 2 3e 1 )0.2996(1a 1 1e 1 1a 1 )0.1449(1e 1 3e 1 1a 1 ) 2 A (1e 2 3e 1 )0.1460(1a 2 1a 1 )0.1326(1e 2 1a 1 )0.1218(1e 1 3e 1 1a 1 ) 2 E (1a 1 1e 1 1a 1 )0.2676(1e 2 3e 1 )0.1629(1e 1 3e 1 1a 1 ) 2 E (1e 3 )0.2709(1e 2 1a 1 )0.2062(1a 2 1e 1 ) 4 A (1e 2 1a 1 ) 2 E (1a 1 3e 2 )0.1547(3e 2 2e 1 )0.1180(1a 2 2e 1 ) 2 A (3e 2 1a 1 )0.2075(1a 1 3e 1 2e 1 )0.1234(2e 2 1a 1 ) 4 A (1a 1 1e 1 3e 1 )0.2103(1a 1 1e 1 2e 1 )0.1532(3e 1 1a 1 2e 1 )0.1352(1e 1 3e 1 2a 1 1 ) 2 E (3e 2 1e 1 )0.1491(1a 1 1e 1 2e 1 )0.1329(1a 1 1e 1 3e 1 ) 4 A (1a 1 3e 1 2e 1 )0.1061(3e 2 2a 1 1 ) 2 E (1a 1 3e 1 2e 1 )0.1303(1a 1 3e 1 2a 1 1 )0.1164(3e 3 ) 4 E (1e 1 3e 1 2e 1 )0.1693(1a 1 1e 1 2e 1 )0.1277(1a 1 1e 1 3e 1 ) 2 A (3e 2 1e 1 )0.1505(1e 1 3e 2 )0.1147(1a 1 1a 1 2a 1 1 ) 2 A (1a 1 1e 1 2e 1 )0.2089(1a 1 1e 1 3e 1 )0.1501(1e 1 3e 1 2a 1 1 )0.1256(3e 1 1a 1 2e 1 ) 2 A (3e 2 2e 1 )0.2167(1a 2 2a 1 1 )0.1247(1a 1 3e 2 ) 2 A (1a 1 1e 1 2e 1 )0.1845(1a 1 1e 1 3e 1 )0.1539(1e 1 3e 1 2a 1 1 )0.1063(3e 2 1a 1 ) 2 E (1e 1 3e 1 2e 1 )0.1539(1a 1 1e 1 2e 1 )0.1049(1a 1 1e 1 3e 1 ) 4 E (1a 1 3e 1 2e 1 )0.1993(1a 1 3e 1 2a 1 1 )0.1831(3e 2 2e 1 ) 2 E (3e 2 1e 1 )0.1192(1e 1 3e 1 2e 1 )0.1107(1a 1 1e 1 2a 1 1 ) 2 A (1e 2 3e 1 )0.2298(1e 2 2e 1 )0.1007(1e 1 3e 1 1a 1 ) 4 A (1e 1 3e 1 2e 1 )0.1322(1a 1 1a 1 2a 1 1 )0.1238(1a 1 1e 1 2e 1 ) 4 E (1e 1 3e 1 2e 1 )0.1514(1a 1 1e 1 2a 1 1 ) 4 E (1e 2 2e 1 )0.1780(1e 2 3e 1 )0.1681(1e 1 3e 1 1a 1 )0.1515(1e 1 1a 1 2a 1 1 ) 2 E (1e 1 3e 1 2e 1 )0.1725(1a 1 1e 1 2a 1 1 )0.1149(3e 1 1a 1 2e 1 ) 2 A (1e 1 3e 1 2e 1 )0.1458(1a 1 1a 1 2a 1 1 )0.1027(1a 1 1e 1 2e 1 ) 2 E (1e 2 2e 1 )0.1673(1e 2 3e 1 )0.1101(1e 1 1a 1 2a 1 1 ) 2 E (1e 2 2e 1 )0.1819(1e 2 3e 1 )0.1310(1e 1 1a 1 2a 1 1 )0.1006(1e 1 3e 1 1a 1 ) 4 A (1e 1 1a 1 2e 1 )0.2820(1e 1 3e 1 1a 1 )0.2808(1e 2 2a 1 1 ) 2 A (1e 1 1a 1 2e 1 )0.2786(1e 2 2a 1 1 )0.2571(1e 1 3e 1 1a 1 ) 4 A (1a 1 3e 1 2e 1 )0.2919(3e 1 2e 1 2a 1 1 ) 2 A (3e 2 2a 1 1 )0.1926(1a 1 3e 1 2e 1 )0.1339(1a 1 2e 2 )0.1188(3e 1 2e 1 2a 1 1 ) 2 E (3e 2 2e 1 )0.1139(1a 1 3e 1 2a 1 1 ) 2 E (3e 2 2e 1 )0.1265(1a 1 2e 1 2a 1 1 )0.1048(3e 1 2e 1 2a 1 1 ) 2 A (1e 1 3e 1 2e 1 )0.1062(1e 1 2e 1 2a 1 1 ) 4 E (1e 1 3e 1 2e 1 )0.1419(1e 1 3e 1 2a 1 1 )0.1053(1e 1 2e 2 )0.1048(1e 1 2e 1 2a 1 1 ) 2 E (1e 1 3e 1 2e 1 )0.1112(1e 1 3e 1 2a 1 1 ) 4 E (3e 1 2e 1 2a 1 1 )0.1727(1a 1 3e 1 2a 1 1 )0.1675(3e 2 2e 1 )0.1616(1a 1 2e 1 2a 1 1 ) (3e 1 2e 2 ) 2 A (3e 2 2e 1 )0.1498(2a 1 2 1a 1 )0.1100(3e 1 2e 2 )0.1100(2e 2 3e 1 ) 2 E (1e 1 3e 1 2e 1 ) 4 A (1e 1 3e 1 2a 1 1 )0.2712(1e 1 2e 1 2a 1 1 )0.1298(3e 1 1a 1 2e 1 ) 2 A (1e 1 2e 1 2a 1 1 )0.2596(1e 1 3e 1 2a 1 1 )0.1427(3e 1 1a 1 2e 1 ) 4 A (1a 1 1e 1 3e 1 )0.1985(1a 1 1e 1 2e 1 ) 2 E (1a 2 1e 1 )0.1354(1e 1 3e 1 2e 1 ) 2 E (1e 2 3e 1 )0.2416(1a 1 1e 2 ) 4 A (1e 1 3e 1 1a 1 )0.3237(1a 1 1e 2 ) 2 A (1a 2 1a 1 )0.2081(1a 1 1e 1 3e 1 ) 4 E (1a 1 1e 1 3e 1 )0.2345(1a 1 3e 1 1a 1 )0.1052(1e 1 3e 1 2e 1 ) 4 E (1a 1 1e 1 1a 1 )0.2703(1e 2 3e 1 ) 2 A (1e 1 3e 1 1a 1 )0.1895(1e 2 3e 1 )0.1732(1e 2 2e 1 )0.1384(1a 1 1e 2 ) 2 E (3e 2 2e 1 )0.1159(3e 1 2e 1 2a 1 1 ) 4 A (3e 1 2e 1 2a 1 1 )0.2469(3e 2 2a 1 1 )0.1960(2e 2 2a 1 1 )

12 9542 Handke et al.: Molecular triple ionization spectra TABLE IV. Continued. State TIP ps Configuration 2 A (1a 1 1e 1 3e 1 ) 4 E (1e 1 3e 2 )0.1439(1e 1 3e 1 2e 1 )0.1251(1a 1 3e 1 1a 1 )0.1038(1a 1 1e 1 2e 1 ) 2 A (1e 2 1a 1 )0.1829(1e 1 3e 1 1a 1 )0.1431(1e 2 3e 1 )0.1279(1a 2 1a 1 ) 2 E e 22a E (3e 1 2e 1 2a 1 1 )0.2009(2a 1 2 3e 1 )0.1360(2e 2 3e 1 ) 2 E (1a 1 2e 1 2a 1 1 ) 2 A (2a 1 2 1a 1 )0.1343(2e 2 1a 1 ) 2 E (2e 2 2a 1 1 )0.1758(2e 3 ) 2 A (2e 2 2a 1 1 )0.2026(3e 1 2e 1 2a 1 1 ) 2 A (2e 2 2a 1 1 )0.1164(3e 1 2e 1 2a 1 1 ) 2 E (2a 1 2 2e 1 )0.1217(2e 2 2a 1 1 ) 4 A (1a 1 3e 1 1e 1 )0.1037(1a 1 2e 1 1e 1 ) 2 E (1a 2 1e 1 )0.1428(3e 2 1e 1 )0.1055(1a 1 3e 1 1e 1 ) 4 E (1a 1 1e 1 1e 1 )0.2081(1e 1 3e 1 1e 1 ) 4 A (1a 1 1e 1 1e 1 ) 2 A (1a 1 1e 1 1e 1 )0.1462(3e 1 1a 1 1e 1 )0.1399(1e 1 3e 1 1a 1 1 ) (1a 1 1e 1 1e 1 ) 4 E (1a 1 3e 1 1e 1 )0.1715(1a 1 3e 1 1a 1 1 )0.1307(3e 2 1e 1 ) 2 E (1e 1 3e 1 1e 1 )0.2054(1a 1 1e 1 1e 1 ) 2 A (1a 2 1a 1 1 )0.1735(3e 2 1e 1 )0.1556(1a 1 3e 1 1e 1 ) 2 A (1e 1 3e 1 1e 1 )0.1521(1a 1 1e 1 1e 1 )0.1185(1a 1 1a 1 1a 1 1 ) 4 A (1e 1 3e 1 1e 1 )0.1517(1a 1 1e 1 1e 1 )0.1155(1a 1 1a 1 1a 1 1 ) 2 A (1a 1 3e 1 1e 1 )0.1267(1a 1 2e 1 1e 1 )0.1209(3e 2 1a 1 1 ) 2 E (1a 1 3e 1 1e 1 )0.1808(1a 1 3e 1 1a 1 1 )0.1397(3e 2 1e 1 ) 2 A (2e 1 2a 1 1 1e 1 ) 4 A (1a 1 1e 2 )0.1896(3e 1 1e 1 1a 1 1 ) 2 E E E (1a 1 1e 1 1a 1 1 )0.1932(3e 1 1e 2 )0.1053(3e 1 1e 1 1a 1 1 ) 4 E (1e 1 1e 2 )0.1530(1e 1 1e 1 1a 1 1 ) 2 A (1a 1 1e 2 )0.1635(3e 1 1e 1 1a 1 1 ) whereas the second type shows a two-site localization with two holes on the same and the third hole on another fluorine atom. In Table V the results of the population analysis are given for the lowest triply ionized states the averages of groups with similar populations are shown. The states of the first type are divided into four groups of which the last, containing only the two states with TIPs of and ev, is already in the energy region of the states with a pole strength of about 0.8. This grouping can be understood from the 3h composition of the ADC eigenvectors of these states. Because of the strong mixing of configurations an investigation in terms of single orbitals is inappropriate. Combining the four outermost orbitals, which are essentially symmetry adapted combinations of the fluorine lone pairs, into one group and the remaining orbitals into a second group, the situation becomes clearer. In Fig. 8 the contributions of the fluorine lone pairs to the total 3h compositions of the ADC eigenvectors of the states up to a triple ionization energy of 150 ev are shown. These contributions are given by n r c 2 r, 18 r where the sum is over all 3h configurations, c r is the coefficient of the 3h configuration r in the ADC eigenvector, and n r is the number of fluorine lone pairs in the configuration r. Therefore 3 means that all 3h configurations which contribute to the corresponding state involve only the fluorine lone pairs. Figure 8 shows that the four groups of three-site states mentioned above are characterized by a different magnitude of the contributions of outer orbitals. For all states in the first group is greater than 2.5 and reaches the maximum of 3. Then is decreasing and reaches a minimum of about 1 in the fourth group. The tricationic states are further differentiated in Fig. 8 in the three-site F 1 F 1 F 1, the two-site F 2 F 1, the one-site F 3, and the remaining states which are all delocalized. This shows that the first two-site states are lying between the third and fourth group of three-site states, having again a contribution of outer orbitals greater than 2.5. Also for these states then decreases with increasing triple ionization energy. In this case, however, the distribution of the values of are more continuous than for the three-site states and therefore no arrangement in different groups is visible in the spectrum. Starting with a TIP of ev there is again a group of three-site states which now have a maximal value of of about 2. These states are the first in which one of the innermost orbitals the 1e and 1a 1 orbitals contributes to the 3h composition of the ADC eigenvector. The quite large energy separation between the 1e and the following 2a 1 orbital of about 21 ev is reflected in the energy separation between the lowest three-site states and those in-

13 Handke et al.: Molecular triple ionization spectra 9543 TABLE V. Results of the three-hole population analysis for BF 3. The average over groups of similar states is given and the energy ranges which comprise the states are indicated. The first line for each group shows the average value of the different contributions and the second line the corresponding standard deviations. The last two lines show the results for two individual states. Their TIPs are in the energy range of the last group, but their character is different from the other states of this group and were therefore considered separately. Energy interval B 3 F 3 F 2 F 1 F 1 F 1 F 1 F 2 B 1 B 2 F 1 F 1 F 1 B 1 Total volving the 1e orbital. Correspondingly, above 100 ev triple ionization energy there is a large group of two-site states which are characterized by 3h compositions with one hole in one of the two innermost orbitals. At about the same energy there are the first F 3 one-site states. These have again a high value of greater than 2, indicating that they FIG. 8. The effective number of fluorine lone pair orbitals see Eq. 18 contributing to the 3h configurations of the ADC eigenvectors of BF 3 for the F 1 F 1 F 1 three-site a, the F 2 F 1 two-site b, the F 3 one-site c localized states and for the remaining delocalized states d. are of the same type, concerning the 3h composition, as the lowest three- and two-site states. The last groups of three-, two-, and one-site localized states visible in Fig. 8, with values of of about 1, are characterized by 3h compositions with one more hole in one of the innermost orbitals. The first delocalized states are found at a triple ionization energy of about 100 ev. Their values of are distributed without an obvious regularity and are always less than 2. The latter shows that there are no delocalized states with all three holes in the fluorine lone pair orbitals. 4. Hole repulsion energies and relaxation effects Given the pronounced hole localization effects found to take place in the tricationic fluorides, it seems instructive to analyze the ADC results in comparison to what can be deduced from a simple electrostatic model of the trications. In order to carry this out, we must first separate, as far as possible, the energy contributions corresponding to hole repulsion and relaxation for the various groups of states with different hole localization patterns. This can be achieved by a calculation of the triple ionization spectrum within the 3h space only, because the mixing of 3h configurations allows for the localization of the holes but relaxation requires the 4h1p configurations. For BF 3 we obtain from such a calculation the centers of the three groups of states with different localization at the following triple ionization energies: threesite F 1 F 1 F 1 states at about 80.5 ev, two-site F 2 F 1 states at about 95.5 ev, and one-site F 3 states at about ev. In a simple point-charge picture the repulsion of two holes localized at different fluorine atoms is given by the inverse distance between them, which, for the BF 3 molecule, amounts to 6.4 ev. The repulsion of two or three holes localized on one atom cannot be calculated in a simple pointcharge model. This information can, however, be obtained from the separation in energy of the groups of states with different localization together with the value for the repulsion of two holes on different atoms obtained from the pointcharge model. The energy separation of the three-site and the

14 9544 Handke et al.: Molecular triple ionization spectra one-site states, for example, is about 43 ev; since the repulsion energy in the three-site states can be calculated in the point-charge model to be about 19 ev, one obtains a repulsion energy of the three holes at the same atom of about 62 ev. Similarly, the hole hole repulsion of two holes at the same atom is calculated from the separation in energy of the two-site and the three-site states, giving 21 ev see also Ref. 22. These figures are found to hold precisely also for the BeF 2 molecule. In this case the point-charge model repulsion of two holes on different atoms is 5.1 ev. If we assume that the repulsion energies of two and three holes on the same fluorine atom are the same as for the BF 3 molecule namely 21 and 62 ev, respectively we can now calculate the energy separation of the one-site and two-site states, yielding a value of about 31 ev. This value compares very well with the result of the ADC calculation for BeF 2 in the 3h configuration space from which an energy separation of the one-site and two-site states of about 30.5 ev is obtained. The comparison of the results of the calculation within the space of 3h configurations only and of the full secondorder ADC calculation allows us to estimate the relaxation energy. For BF 3, relaxation energies of about 7, 11.5, and 21 ev are found for the three-site, two-site, and one-site states, respectively. The relaxation energy is quadratic in the hole charge so that the ratios of the relaxation energies of the states with different hole localization in a point charge model are 3:5:9 3 charges of magnitude 1 at different atoms for the three-site states, a charge of magnitude 2 at one atom and one of magnitude 1 at another for two-site states, and a charge of magnitude 3 at one atom for the one-site states. The ratios obtained from the ADC calculations for BF 3 correspond indeed very well to this simple picture, further underlining the strong hole localization. For BeF 2 the relaxation energies are found to be about 10 and 15 ev for the two-site and one-site states, respectively. The ratio of these values corresponds less accurately to the simple point-charge model ratio of 5:9. This indicates that the hole localization is stronger in the BF 3 molecule than in BeF 2. We conclude the discussion of the triply ionized states of LiF, BeF 2, and BF 3 with a comparison of the spectra of the one-site F 3 localized states of these molecules. This analysis can be useful for a qualitative interpretation of the shakeoff satellite contributions to the corresponding Auger spectra. 29 Figure 9 shows the bar spectra of the F 3 terms of the three-hole population analysis together with a convolution of the bar spectra with Gaussian functions of full width at half-maximum of 1 ev. The spectra of LiF and BeF 2 are very similar. The main difference is a shift in energy of about 15 ev due to the symmetry-breaking localization effects in BeF 2 leading to a large energy split between the one-site and the two-site localized tricationic states. In both spectra the individual features are due to the strong contributions of a few states. This situation changes drastically for the BF 3 molecule. Here a much larger number of states contributes and therefore the shape of the spectrum is produced by the sum of very many rather small contributions. FIG. 9. The F 3 one-site terms for LiF, BeF 2, and BF 3 as a bar spectrum and convoluted with Gaussian functions of full width at half-maximum of 1.0 ev. IV. SUMMARY AND CONCLUSIONS The newly implemented second-order ADC approximation of the three-particle propagator has been used for the theoretical study of triple ionization of molecules. This method enables the accurate calculation of a very large number of triple ionization energies and corresponding transition amplitudes. To analyze the ADC eigenvectors the atomic three-hole population analysis has been developed. From this analysis information is obtained about the charge distribution and space localization of electron vacancies in the dense manifold of tricationic states. In exemplary applications it has been demonstrated that the calculation of a large number of triply ionized states and the investigation of localization effects using the three-hole population analysis are very helpful for the theoretical understanding of the triple ionization of molecules and for the interpretation of the rapidly growing body of experimental data on molecular trications. The application of our method to the triple ionization cross section of carbon monoxide has given strong evidence that a vast number of molecular tricationic states are produced in experiments with high energy projectiles. An interpretation of available experimental data on the base of only a few low-lying triply ionized states is therefore not reliable. Shake-off processes in Auger spectroscopy lead to triply ionized final states. On the example of the Auger spectrum of LiF it has been shown that with an accu-

15 Handke et al.: Molecular triple ionization spectra 9545 rate calculation of the triple ionization potentials and an estimate of the Auger intensities based on the three-hole population analysis the shake-off satellite bands can be reproduced with satisfactory accuracy. From the study of the triple ionization of the fluorides LiF, BeF 2, and BF 3 it becomes evident how the number of relevant tricationic states grows very rapidly with the size of the molecule. This rapid growth of the density of states is accompanied by an increasing mixing of configurations. It is clear from these results that conventional self-consistent field methods cannot be used in these studies, while for more adequate CI methods the numerical effort becomes prohibitive because of the large number of states which has to be calculated. In spite of the very large number of triply ionized states and the complexity of the corresponding spectra the overall pattern of the distribution and grouping of states can be explained in simple terms. With help of the atomic three-hole population analysis the charge distribution over the trication can be studied. The population analysis yields a decomposition of the total three-hole pole strength in contributions describing the localization of the three holes on the same, on two, or on three different atoms. Because of the different hole repulsion energies in the differently localized states, this information enables an explanation of the triple ionization spectra in terms of the 3h composition of the calculated states. In the experimental observations, the effects of nuclear dynamics on the triply ionized states going beyond the analysis of vertical transitions can of course be relevant. This is in general an extremely difficult subject for theoretical analysis, as it may require detailed knowledge of the whole potential energy surfaces for an enormous number of states and of the nonadiabatic couplings among them. In the case of decay spectroscopies Auger these effects are, for the fluoride compounds discussed in the present work, uniform and of minor importance. They can in general be effectively estimated by the methods developed in Ref. 33 see for example Ref. 14. The total dissociation cross section of trications has been found to be largely governed, instead, by the very dense energy distribution of triply ionized states. We hope to have shown that the methods presented here have a wide range of applicability and constitute a suitable instrument for the investigation and understanding of molecular triple ionization processes. ACKNOWLEDGMENTS The authors wish to thank the Vigoni program between C.R.U.I., Italy and D.A.A.D., Germany for traveling funds. 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